ACJC 2025 JC2 H2 Prelim Paper 2 QP

ANGLO-CHINESE JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION Higher 2 MATHEMATICS 9758/02 Paper 2 29 August 2025 QUESTION PAPER 3 hr Additional Materials: Printed Answer Booklet List of Formulae and Results (MF27) READ THESE INSTRUCTIONS FIRST Answer all questions. Write your answers on the Printed Answer Booklet . Follow the instructions on the front cover of the answer booklet. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. You are expected to use an approved graphing calculator. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphing calculator are not allowed in a question, you are required to present the mathematical steps using mathematical notations and not calculator commands. You must show all necessary working clearly. The number of marks is given in brackets [ ] at the end of each question or part question. _________________________________________________________________________________ This document consists of 8 printed pages. [Turn over

2 ANGLO-CHINESE JUNIOR COLLEGE 2025 H2 MATHEMATICS 9758/02 Section A: Pure Mathematics [40 marks] 1 It is given that 3 + 3OA += i j k and 5 4 3OB −+= i j k . Find the position vector of a point R on line OB such that AR is perpendicular to OB. Hence find the position vector of the point A, the reflection of the point A in the line OB. [4] 2 It is given that 2 1f( ) 14 x x = − where 11 22 x−   . (a) Using standard series from the List of Formula e (MF27), find the Maclaurin’s expansion of f( )x , up to and including the term in 6x . [2] (b) Hence find the first four non-zero terms of the Maclaurin series for 1sin 2 x− . [4] 3 (a) For this question, you may use these results: ( )( )2 1 1 2 1 6 n r n n nr = ++= and ( )3 22 1 4 1 n r nnr = += . (i) Show that ( ) ( )( )( )2 1 1 2 3 51 12 n r n n n nrr = + + ++= . [3] (ii) Hence find ( )( ) 1 2 5 2 3 n r r r − = + + in terms of n. [3] (b) The sequence 1 2 3, , ,u u u is defined by 1 2u = , 1 1 ,11 n n un u + =− . Find the values of 2u , 3u and 4u . Hence find the value of 2025u . [3]

3 ANGLO-CHINESE JUNIOR COLLEGE 2025 H2 MATHEMATICS 9758/02 [Turn over 4 The function f is defined by 2 1f: 65x xx++ , for 3x− , 1x− . (a) Find ( ) 1f x− . [3] (b) Find algebraically the range of f. [4] The function g is defined by g : e xx , for x . (c) Find the exact range of gf. [2] 5 (a) By using the substitution tanxa = , show that 2 2 2 2 11 d ln xxc ax x a x a a =+ + + +   , where 0x and 0a . [4] (b) At time t in a chemical reaction, x kg of substance X and y kg of substance Y are present. Initi